Subgroup ($H$) information
| Description: | $C_3\times D_{42}$ |
| Order: | \(252\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 7 \) |
| Index: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(42\)\(\medspace = 2 \cdot 3 \cdot 7 \) |
| Generators: |
$a^{2}c^{21}, b^{2}, c^{14}, b^{3}, c^{6}$
|
| Derived length: | $2$ |
The subgroup is normal, a semidirect factor, nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), and an A-group.
Ambient group ($G$) information
| Description: | $D_{42}.D_6$ |
| Order: | \(1008\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 7 \) |
| Exponent: | \(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group.
Quotient group ($Q$) structure
| Description: | $C_4$ |
| Order: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Automorphism Group: | $C_2$, of order \(2\) |
| Outer Automorphisms: | $C_2$, of order \(2\) |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group) and a $p$-group.
Automorphism information
While the subgroup $H$ is not characteristic, the stabilizer $S$ of $H$ in the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : S \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphisms $\operatorname{Inn}(G) \cap S$ is the Weyl group $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{21}.C_6^2.C_2^6$ |
| $\operatorname{Aut}(H)$ | $C_2\times D_6\times F_7$, of order \(1008\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 7 \) |
| $\operatorname{res}(S)$ | $C_2\times D_6\times F_7$, of order \(1008\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 7 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| $W$ | $D_{42}$, of order \(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \) |
Related subgroups
Other information
| Möbius function | $0$ |
| Projective image | $C_6.D_{42}$ |